If the equation of a circle is x 2 + y 2 = r 2 and the equation of a tangent line is y = mx + b,show that r 2 ( 1 + m 2 ) = b 2

Question

If the equation of a circle is       x          2        +      y          2        =      r          2       and the equation of a tangent line is y = mx + b,show that       r          2        (  1  +      m          2        )  =      b          2

cindysnifflesuz · Accepted Answer

Write the equations as a system of equations.      x          2        +      y          2        =      r          2      y = mx + bIf we substitute the second into the first, we get the above, andsince the circle intersects once, only one x value can satisfy thatequation.Now we need to expand:      x          2        +  (  m  x  +  b      )          2        =      r          2            x          2        +      m          2            x          2        +  2  m  b  x  +      b          2        −      r          2        =  0  (  1  +      m          2        )      x          2        +  2  m  b  x  +  (      b          2        −      r          2        )  =  0If this has one solution, then the discriminant must be 0, i.e.  (  2  m  b      )          2        −  4  (  1  +      m          2        )  (      b          2        −      r          2        )  =  0  4      m          2            b          2        −  4      b          2        −  4      m          2            b          2        +  4      r          2        +  4      r          2            m          2        =  0  −  4      b          2        +  4      r          2        +  4      r          2            m          2        =  0  −      b          2        +      r          2        +      r          2            m          2        =  0      r          2        (  1  +      m          2        )  =      b          2

If the equation of a circle is x^2 + y^2 =r^2 and the equation of a tangent line is y = mx + b,show that r^2(1 + m^2) =b^2

Answered question

Answer & Explanation

New Questions in Algebra I